Oscillating dipole quantum number

J. Troe My answer to Prof. Herman is that the high-Stark-field description of the close approach of a dipole to an ion can very well be represented in terms of the relevant quantum numbers. The linear dipole-free rotor quantum numbers j and m are converted to the oscillating dipole quantum number v with the identity v - 2j - m. ... 

An ordered monolayer of molecules having a large dynamical dipole moment must not be regarded as an ensemble of individual oscillators but a strongly coupled system, the vibrational excitations being collective modes (phonons) for which the wavevector q is a good quantum number. The dispersion of the mode for CO/Cu(100) in the c(2 x 2) structure has been measured by off-specular EELS, while the infrared radiation of course only excites the q = 0 mode. 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

Simultaneously with the oscillations with respect to the internuclear axis, a diatomic molecule rotates as a whole with a frequency ft (not to be confused with the quantum number of the projection of the total momentum, as in Section 1.2) around an axis which is perpendicular to the direction of the internuclear axis, exhibiting a total angular momentum J see Fig. 1.3. Hence in the laboratory coordinate system, with respect to which the light wave possesses a certain polarization, electrons participate simultaneously in two types of motion dipole oscillations with respect to the internuclear axis, and rotation of the molecule as a whole. 

Deutschbein ". Most of the intraconfigurational 4f transitions in the RE ions, however, cannot be accounted for by the magnetic dipole mechanism, not only because the predicted oscillator strengths are in general smaller than 10 , but also due to the restrictive selection rules on the total angular momentum 7, A7 =0, 1 (0 0 excluded), as far as 7 is considered to be a good quantum number. 

In addition to the selection rules restricting the changes in the quantum numbers, the presence or absence of a dipole moment in the molecule imposes a restriction on the appearance of lines and bands in the spectrum. If the transition between one vibratio.nal or rotational state to another is to produce emission or absorption of radiation the vibration or rotation must be accompanied by an oscillation in the magnitude of the dipole moment of the molecule. 

To employ the one-dimensional treatment of infrared transitions efficiently, it is worthwhile to consider some more complex situations. The general procedure, however, remains the same as the one outlined previously for two interacting bonds. We now attach a bond (dipole) operator to each one-dimensional internal degree of freedom of the molecule. Such operators will change the local vibrational quantum number by 0 1, 2, units leading to exponentially decreasing expectation values of the infrared transition O u. The total dipole moment operator for n oscillator is given by... 

Of much greater importance than the rotation spectra are the spectra of diatomic molecules corresponding to simultaneous changes of both the vibrational and the rotational quantum number, the vibration-rotation spectra. In analogy with the classical result they are present only in molecules for which the oscillation of the nuclei is accompanied by an oscillation of the dipole moment so that homonuclear molecules have no vibration-rotation spectra. The quantum... 

The reader can verify that the photon which has the proper energy to increase the quantum number by two has a frequency far higher than the initial or final classical rotational frequency involved in such a transition. In fact, it has a frequency a little higher than twice the initial classical rotational frequency but there is no dipole moment component oscillating at this frequency. There is no anharmonicity in rotation. This corresponds to a quantum mechanically forbidden transition. 

Selection rules for spectroscopic experiments are derived from time-dependent perturbation theory. Transitions are allowed if the integral of the perturbing Hamiltonian and initial and final stationary states is nonzero. For the harmonic oscillator, the allowed transitions are those for which the n quantum number changes by 1, provided that there exists a dipole that varies linearly with the separation distance. (N2 has a zero dipole because of its symmetry, and there is no linear variation with distance.) For the rigid rotator, the selection rules are that the allowed transitions are a change of 1 in the / quantum number, provided that there is a nonzero permanent dipole, po-... 

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